Dedekind-Finite Rings

Q: Does Dedekind finiteness imply invariant basis number?

Not by itself. IBN follows from *stable* finiteness, that is, Dedekind finiteness of every M_n(R), and Shepherdson's example shows the two conditions differ. For semilocal rings both hold, since M_n(R) is again semilocal.

Q: How does Bass' Theorem generalise (20.8)?

Set B = 0: the hypothesis becomes Ra = R, that is, a has a left inverse, and the conclusion is that a itself is a unit. That is precisely Dedekind finiteness. Nonzero B gives freedom to move a within a coset before demanding invertibility.

Executive Summary

In a noncommutative ring a left inverse need not be a right inverse. The rings where the pathology cannot occur are the Dedekind-finite ones: $a b = 1 \Rightarrow b a = 1$ . Equivalently, the free module $R_{R}$ is not isomorphic to a proper direct summand of itself — a ring-theoretic version of "a finite set is not equinumerous with a proper subset", which is where the name comes from.

Semilocal rings are Dedekind-finite, by a two-step argument: semisimple rings are, and the property lifts along $R \to R / rad R$ . Bass' Theorem $(20.9)$ then upgrades this substantially: instead of merely asserting that a left-invertible element is a unit, it produces a unit inside a prescribed coset.

$a b = 1 \Rightarrow b a = 1$ Definition

$(20.8)$ Semilocal implies it

$(20.9)$ Bass' refinement

$𝔅 = 0$ Special case recovering

(20.8)

Overview

Take $V$ a vector space with countable basis $e_{1}, e_{2}, \dots$ over a field $k$ and let $R = {End}_{k} (V)$ . Let $b$ be the shift $e_{i} \mapsto e_{i + 1}$ and $a$ the backward shift $e_{1} \mapsto 0$ , $e_{i} \mapsto e_{i - 1}$ for $i \geq 2$ . Then $a b = 1$ while $b a$ kills $e_{1}$ , so $b a \neq 1$ . This one example is the whole subject: it is why Dedekind finiteness must be assumed or proved, never taken for granted.

Dedekind finiteness is self-dual — the condition $a b = 1 \Rightarrow b a = 1$ is unchanged by swapping the roles of $a$ and $b$ — so there is no left or right version, and it can be tested on right modules or left modules indifferently.

Bass discovered the sharper statement while building algebraic K-theory. His theorem says that in a semilocal ring, whenever $R a$ together with a left ideal $𝔅$ generates the whole ring, the affine translate $a + 𝔅$ already meets the unit group. Taking $𝔅 = 0$ gives Dedekind finiteness back; taking $𝔅$ principal gives the stable range condition of the next page.

Learning Objectives

State Dedekind finiteness and prove the equivalence with $R_{R} ≅ R_{R} \oplus M \Rightarrow M = 0$ .
Prove that $R$ is Dedekind-finite whenever $R / rad R$ is.
Prove that semisimple rings — and more generally rings of finite length — are Dedekind-finite.
Deduce $(20.8)$ : every semilocal ring is Dedekind-finite.
Reconstruct both proofs of Bass' Theorem $(20.9)$ .
Separate Dedekind-finite from stably finite with a correct citation.

Definitions

DefinitionDedekind-finite

A ring $R$ is Dedekind-finite if for all $a, b \in R$ , $a b = 1$ implies $b a = 1$ . Synonyms in the literature: directly finite, von Neumann finite, inverse symmetric.

Left-invertible: a is left-invertible if ba = 1 for some b. In a Dedekind-finite ring left-invertible, right-invertible and invertible coincide.
Stably finite: M_n(R) is Dedekind-finite for every n. Strictly stronger than Dedekind-finite.
$U (R)$: The group of two-sided units of R.
IBN: Invariant basis number: R^n isomorphic to R^m forces n = m. Implied by stable finiteness.

Every commutative ring is Dedekind-finite; the notion has content only in the noncommutative setting.

Core Concepts

The idempotent produced by a one-sided inverse

Suppose $a b = 1$ . Then $e : = b a$ satisfies $e^{2} = b (a b) a = b a = e$ , so $e$ is idempotent and $R_{R} = e R \oplus (1 - e) R$ . The map $x \mapsto b x$ is an injective right $R$ -module endomorphism of $R$ — injective because $a (b x) = x$ — with image $b R = b a R = e R$ . Hence $e R ≅ R_{R}$ and

R_{R} ≅ R_{R} \oplus (1 - e) R .

(20.8a)

A failure of Dedekind finiteness is exactly a nonzero complement $(1 - e) R$ .

So $b a = 1$ if and only if $e = 1$ , if and only if $(1 - e) R = 0$ . Dedekind finiteness is the statement that the free module of rank one is directly finite.

Two sources of the property

Source A

Finite length

An injective endomorphism of a module of finite length is surjective. Applying this to $x \mapsto b x$ on $R_{R}$ gives $b x = 1$ for some $x$ , whence $b$ is a unit and $b a = 1$ . This covers semisimple rings and, more generally, left or right artinian rings.

Source B

No infinite orthogonal idempotents

If $a b = 1$ and $b a \neq 1$ , the elements $e_{i} = b^{i} (1 - b a) a^{i}$ are nonzero, pairwise orthogonal idempotents. A ring admitting no infinite orthogonal family — for instance any left or right noetherian ring — is therefore Dedekind-finite.

Why the radical is transparent

Units are detected modulo the radical: $u \in U (R)$ if and only if $\bar{u} \in U (R / rad R)$ , because $1 + rad R \subseteq U (R)$ . That makes Dedekind finiteness a property of the radical quotient alone, which is precisely what makes the semilocal hypothesis bite.

Key Results

Lemma(4.8)Dedekind finiteness lifts across the radical

Let $R$ be any ring. If $R / rad R$ is Dedekind-finite then so is $R$ .

Proof

Let $a b = 1$ in $R$ and write $\bar{\cdot}$ for reduction modulo $J = rad R$ . Then $\bar{a} \bar{b} = 1$ , so by hypothesis $\bar{b} \bar{a} = 1$ , i.e. $1 - b a \in J$ . Since every element of $1 + J$ is a unit, $b a \in U (R)$ . But $b a$ is idempotent, as computed above, and a unit idempotent equals $1$ : multiplying $e^{2} = e$ by $e^{- 1}$ gives $e = 1$ . Hence $b a = 1$ . $□$

LemmaSemisimple rings are Dedekind-finite

If $R$ is semisimple then $R$ is Dedekind-finite.

Proof

A semisimple ring has finite length as a right module over itself. Suppose $a b = 1$ . The map $ϕ : R_{R} \to R_{R}$ , $ϕ (x) = b x$ , is a right $R$ -module homomorphism, and it is injective because $a ϕ (x) = a b x = x$ . An injective endomorphism of a module of finite length is surjective, so $b x = 1$ for some $x \in R$ . Then $b$ has a left inverse $a$ and a right inverse $x$ , so $b \in U (R)$ and $a = a (b x) = (a b) x = x$ . Therefore $b a = b x = 1$ . $□$

Proposition(20.8)Semilocal rings are Dedekind-finite

Every semilocal ring $R$ is Dedekind-finite.

Proof

By definition $R / rad R$ is semisimple, hence Dedekind-finite by the preceding lemma. Now apply $(4.8)$ . $□$

Theorem(20.9)Bass' Theorem

Let $R$ be a semilocal ring, let $a \in R$ , and let $𝔅$ be a left ideal of $R$ with

R a + 𝔅 = R .

Then the coset $a + 𝔅$ contains a unit of $R$ .

With $𝔅 = 0$ the hypothesis says $a$ has a left inverse and the conclusion says $a$ is a unit, recovering $(20.8)$ .

ProofBass' original argument

Reduction to the semisimple case. An element of $R$ is a unit if and only if its image in $R / rad R$ is a unit. The hypothesis $R a + 𝔅 = R$ passes to the quotient, and a unit found in the image of $a + 𝔅$ lifts to a unit of $R$ lying in $a + 𝔅$ . So assume $R$ is semisimple. Since units, left ideals and the hypothesis all decompose over a finite direct product, we may further assume $R$ is simple artinian, and by Wedderburn–Artin $R = End (V_{D})$ for a finite-dimensional right vector space $V$ over a division ring $D$ .

**The subspace attached to $𝔅$ .** Put $W = {v \in V : 𝔅 v = 0}$ . For $R = End (V_{D})$ with $\dim_{D} V$ finite, every left ideal is the annihilator of a subspace, and $𝔅 = ann (W) = {f \in R : f (W) = 0}$ .

** $a$ is injective on $W$ .** Write $1 = r a + b$ with $r \in R$ , $b \in 𝔅$ . If $w \in W$ and $a (w) = 0$ then $w = (r a + b) (w) = r (a (w)) + b (w) = 0$ , since $b (W) = 0$ . Hence $a |_{W} : W \to a (W)$ is a $D$ -isomorphism.

Extending to an automorphism. Because $\dim_{D} W = \dim_{D} a (W)$ and $V$ is finite-dimensional, the complements of $W$ and of $a (W)$ have equal dimension. Choose any $D$ -isomorphism between those complements and combine it with $a |_{W}$ to obtain $f \in Aut (V_{D}) = U (R)$ with $f (w) = a (w)$ for all $w \in W$ .

Conclusion. $f - a$ vanishes on $W$ , so $f - a \in ann (W) = 𝔅$ , and $f = a + (f - a) \in a + 𝔅$ is a unit. $□$

ProofSwan's argument, free of Wedderburn–Artin

Reduce to $R$ semisimple as before. Choose a left ideal $𝔅'$ with $𝔅 = (𝔅 \cap R a) \oplus 𝔅'$ ; then $R = R a + 𝔅 = R a \oplus 𝔅'$ , and since $a + 𝔅' \subseteq a + 𝔅$ we may replace $𝔅$ by $𝔅'$ and assume $R = R a \oplus 𝔅$ .

Consider the exact sequence of left $R$ -modules $0 \to K \to R \overset{f}{⟶} R a \to 0$ with $f (r) = r a$ and $K = \ker f$ . It splits, so there is $g : R \to K$ with $(f, g) : R \to R a \oplus K$ an isomorphism. Comparing with $R = R a \oplus 𝔅$ and cancelling the common summand $R a$ — legitimate for semisimple modules, where isomorphism classes are determined by multiplicities of simples — yields an isomorphism $θ : K \to 𝔅$ .

The composite

R \overset{(f, g)}{⟶} R a \oplus K \overset{1 \oplus θ}{⟶} R a \oplus 𝔅 = R

is an isomorphism of left $R$ -modules sending $r \mapsto r a + θ (g (r))$ . A left $R$ -module isomorphism $R \to R$ is right multiplication by the image $u$ of $1$ , and it is bijective exactly when $u \in U (R)$ . Since $u = a + θ (g (1))$ and $θ (g (1)) \in 𝔅$ , the coset $a + 𝔅$ contains the unit $u$ . $□$

RemarkWhy two proofs

The first proof is linear algebra over a division ring and makes the geometric content visible: $a$ is invertible where it needs to be, and the rest of $V$ is repaired by hand. The second uses only splitting and cancellation of semisimple modules, so it generalises to settings where no Wedderburn decomposition is available.

Proof Techniques and Method

How these proofs work, and which move to reuse.

1. Reduce modulo the radicalUnits and the hypothesis

R a + 𝔅 = R

are both visible in

R / rad R

, and units lift. Every unit-theoretic proof for semilocal rings starts here.

2. Split the ring as a moduleOver a semisimple ring every submodule is a direct summand, so

R = R a \oplus 𝔅'

can be arranged. Structural questions become bookkeeping with multiplicities of simple modules.

3. Build the unit as an isomorphismA unit of

R

is the same thing as an automorphism of

R R

. To produce a unit in a coset, construct a module isomorphism whose value at

1

lies in that coset.

4. SpecialiseSet

𝔅 = 0

to recover Dedekind finiteness; set

𝔅 = R b

to obtain left stable range one.

Worked Example

A ring that fails, and why it is not semilocal

Let $V$ have basis $e_{1}, e_{2}, \dots$ over a field $k$ and $R = {End}_{k} (V)$ , acting on the left. Define $b (e_{i}) = e_{i + 1}$ and $a (e_{1}) = 0$ , $a (e_{i}) = e_{i - 1}$ for $i \geq 2$ . Then $a b (e_{i}) = a (e_{i + 1}) = e_{i}$ , so $a b = 1$ ; but $b a (e_{1}) = b (0) = 0 \neq e_{1}$ , so $b a \neq 1$ . Concretely $b a$ is the projection onto the span of $e_{2}, e_{3}, \dots$ , and

R_{R} ≅ R_{R} \oplus (1 - b a) R, (1 - b a) R \neq 0 .

(20.8b)

$R$ absorbs a free summand: the module-theoretic face of $a b = 1 \neq b a$ .

By $(20.8)$ this ring cannot be semilocal — a useful negative test. Indeed $rad R = 0$ while $R$ is far from artinian. The subalgebra generated by $a$ and $b$ is $k ⟨ x, y ⟩ / (x y - 1)$ , which inherits the failure and is the smallest standard example.

Bass' Theorem in a small semilocal ring

Take $R = ℤ / 12$ , semilocal since $ℤ / 12 ≅ ℤ / 4 \times ℤ / 3$ , with $U (R) = {1, 5, 7, 11}$ and $rad R = (6)$ .

Instances of $(20.9)$ in $ℤ / 12$
$a$	$𝔅$	Check $R a + 𝔅 = R$	Coset $a + 𝔅$	Unit found
$2$	$(3)$	$(2) + (3) = (1)$	${2, 5, 8, 11}$	$5$ and $11$
$4$	$(3)$	$(4) + (3) = (1)$	${4, 7, 10}$	$7$
$3$	$(4)$	$(3) + (4) = (1)$	${3, 7, 11}$	$7$ and $11$
$6$	$(0)$	$(6) \neq (1)$ — hypothesis fails	${6}$	none; no contradiction

The last row is the reminder that the hypothesis is doing work: $6$ is a non-unit and the coset $6 + 0$ contains nothing else, but $R \cdot 6 = (6) \neq R$ , so $(20.9)$ does not apply.

Comparison and Classification

Which classes of rings are Dedekind-finite, and why
	Dedekind-finite	Stably finite	Reason
Commutative	yes	yes	$a b = 1$ is symmetric; matrix rings over commutative rings are stably finite via determinants.
Semisimple	yes	yes	Finite length; $M_{n}$ of semisimple is semisimple.
Left or right artinian	yes	yes	Finite length argument; $M_{n} (R)$ is again artinian.
Left or right noetherian	yes	yes	No infinite orthogonal idempotent family; $M_{n} (R)$ is again noetherian.
Local	yes	yes	Semilocal, and $M_{n}$ of a local ring is semilocal by $(20.4)$ .
Semilocal	yes	yes	$(20.8)$ ; $M_{n} (R)$ is semilocal by $(20.4)$ , so $(20.13) (4)$ applies.
Left stable range one	yes	yes	Take $b = 0$ in the definition; $(20.13) (4)$ for the matrix rings.
General ring	no	no	${End}_{k} (V)$ with $\dim V$ infinite.
Dedekind-finite ring	yes	partial	There exist Dedekind-finite $R$ with $M_{2} (R)$ not Dedekind-finite (Shepherdson).

Which classes of rings are Dedekind-finite, and why

"part" records that the implication fails in general although it holds for every named class above it.

Relationship Map

semilocalleft stable range

1

stably finiteDedekind-finiteIBN

Reading left to right: $(20.9)$ , then $(20.13) (4)$ , then the case $n = 1$ , then $(20.13) (2)$ . No arrow reverses. In particular a ring of left stable range one need not be semilocal — an infinite product of fields is unit-regular, hence of stable range one, but has zero radical and no chain condition.

Dedekind finiteness — $a b = 1 \Rightarrow b a = 1$
- Module form — $R_{R} ≅ R_{R} \oplus M \Rightarrow M = 0$
  - No free summand can be absorbed
  - Every surjective endomorphism of $R_{R}$ splits injectively
- Matrix form — $M_{n} (R)$ Dedekind-finite for all $n$ = stably finite
  - Implies invariant basis number
  - Implies every epimorphism $R^{n} \to R^{n}$ is an isomorphism

Applications and Industry Use

Applications here means where this structure is used — inside mathematics and in the engineering and computing disciplines that consume it.

Algebraic K-theory. Bass' Theorem is the base case of the stability theorems for $K_{1}$ : it is why $K_{1}$ of a semilocal ring is generated by units and why relative sequences are exact at the right places.
Operator algebras. Direct finiteness of von Neumann algebras is the algebraic shadow of the type classification; a factor is finite exactly when its underlying ring is Dedekind-finite, which is how type ${II}_{1}$ is separated from type ${II}_{\infty}$ .
Group rings. Kaplansky proved that $k G$ is Dedekind-finite for every group $G$ when $char k = 0$ , using a trace argument; Elek and Szabó extended this to all fields when $G$ is sofic. The general case is open and is a well-known test problem.
Linear systems and modules of states. When a module of signals over a ring of operators can absorb a free summand, feedback constructions that assume finite rank silently fail. Dedekind finiteness of the coefficient ring is the hypothesis that rules this out.
Symbolic computation. Algorithms that invert matrices over a coefficient ring by finding a one-sided inverse are only correct when the coefficient ring is stably finite; over a semilocal ring this is guaranteed by $(20.13) (4)$ .

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

1888Dedekind's finitenessA set is finite when it admits no bijection with a proper subset. The ring-theoretic condition is the same idea applied to $R_{R}$ , and it keeps Dedekind's name.
1936–1940von Neumann's continuous geometriesDirect finiteness is isolated as the algebraic content of finiteness for rings of operators, giving the alternative name von Neumann finite.
1951Shepherdson's separationA Dedekind-finite ring whose $2 \times 2$ matrix ring is not Dedekind-finite, showing that stable finiteness is a genuinely stronger condition.
1964Bass' TheoremIn his foundational K-theory paper Bass proves that semilocal rings have stable range one, in the coset form $(20.9)$ .
1968Swan's proofSwan gives the module-theoretic argument, avoiding Wedderburn–Artin and making the result portable to other settings.
2004Sofic groupsElek and Szabó prove that $k G$ is Dedekind-finite for every field $k$ and every sofic group $G$ , extending Kaplansky's characteristic-zero theorem.

Quick Reference

Definition

a b = 1 \Rightarrow b a = 1

. Self-dual; no left or right version.

Module form

R_{R} ≅ R_{R} \oplus M \Rightarrow M = 0

$(20.8)$ Semilocal

\Rightarrow

Dedekind-finite, via

(4.8)

plus finite length of

R / rad R

$(20.9)$

R

semilocal,

𝔅

a left ideal,

R a + 𝔅 = R

\Rightarrow

(a + 𝔅) \cap U (R) \neq \emptyset

Standard failure

{End}_{k} (V)

\dim_{k} V

infinite; equivalently

k ⟨ x, y ⟩ / (x y - 1)

Sufficient conditions for Dedekind finiteness
Hypothesis	Argument
$R$ commutative	Multiply $a b = 1$ on both sides.
$R$ has finite length as a module over itself	Injective endomorphism of a finite-length module is onto.
$R$ left or right noetherian	No infinite family of nonzero orthogonal idempotents.
$R / rad R$ Dedekind-finite	$(4.8)$ : $1 - b a \in rad R$ forces the idempotent $b a$ to be a unit.
$R$ semilocal	$(20.8)$ .
$R$ has left stable range one	Take $b = 0$ in the stable range condition.

Frequently Asked Questions

Why is it called Dedekind finiteness?

Dedekind defined a set to be finite when it admits no bijection with a proper subset. The ring condition $a b = 1 \Rightarrow b a = 1$ is equivalent to $R_{R}$ admitting no isomorphism with a proper direct summand of itself, so it is the exact module-theoretic transcription.

Does Dedekind finiteness imply invariant basis number?

Not by itself. IBN follows from stable finiteness, that is, Dedekind finiteness of every $M_{n} (R)$ , and Shepherdson's example shows the two conditions differ. For semilocal rings both hold, since $M_{n} (R)$ is again semilocal.

How does Bass' Theorem generalise

(20.8)

Set $𝔅 = 0$ : the hypothesis becomes $R a = R$ , that is, $a$ has a left inverse, and the conclusion is that $a$ itself is a unit. That is precisely Dedekind finiteness. Nonzero $𝔅$ gives freedom to move $a$ within a coset before demanding invertibility.

Why does the proof reduce to a semisimple ring so easily?

Because an element is a unit exactly when its image modulo the radical is, and because both the hypothesis and the conclusion are statements about cosets and units. Nothing about $rad R$ itself is needed — only that the quotient is semisimple.

Is Dedekind finiteness preserved by subrings, quotients or products?

Subrings: yes, trivially, since the condition is universal. Arbitrary direct products: yes, componentwise. Quotients: no — a quotient of a Dedekind-finite ring can fail the condition, since new one-sided inverses may appear.

Is it known whether

k G

is always Dedekind-finite?

Not in general. Kaplansky settled characteristic zero for all groups by a trace argument, and Elek–Szabó settled all characteristics for sofic groups. Whether it holds for every group and every field remains open and is tied to the existence of non-sofic groups.

References

T. Y. Lam, A First Course in Noncommutative Rings, Graduate Texts in Mathematics 131, Springer-Verlag, 1991, §20, (20.8)–(20.9), pp. 313–314, and §4, (4.8).
H. Bass, “K-theory and stable algebra”, Publications Mathématiques de l'IHÉS 22 (1964), 5–60.
R. G. Swan, Algebraic K-Theory, Lecture Notes in Mathematics 76, Springer-Verlag, 1968.
J. C. Shepherdson, “Inverses and zero divisors in matrix rings”, Proceedings of the London Mathematical Society (3) 1 (1951), 71–85.
K. R. Goodearl, Von Neumann Regular Rings, 2nd edition, Krieger, 1991, for direct finiteness, unit-regularity and stable range.
P. M. Cohn, Free Rings and Their Relations, 2nd edition, Academic Press, 1985, for weakly finite rings and invariant basis number.

AI Suggested Questions

Write out Shepherdson's example of a Dedekind-finite ring with $M_{2} (R)$ not Dedekind-finite.
Prove in detail that $e_{i} = b^{i} (1 - b a) a^{i}$ are nonzero orthogonal idempotents when $a b = 1 \neq b a$ .
Give Kaplansky's trace proof that $k G$ is Dedekind-finite in characteristic zero.
Show that a von Neumann regular ring is Dedekind-finite if and only if it is unit-regular.
Derive left stable range one for semilocal rings directly from Bass' Theorem, and identify where the left ideal must be principal.
Which of Dedekind-finite, stably finite and stable range one are Morita invariant?
Construct a Dedekind-finite ring failing invariant basis number, or prove none exists.

Dedekind Finiteness

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

The idempotent produced by a one-sided inverse

Two sources of the property

Finite length

No infinite orthogonal idempotents

Why the radical is transparent

Key Results

Proof Techniques and Method

Worked Example

A ring that fails, and why it is not semilocal

Bass' Theorem in a small semilocal ring

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

References

AI Suggested Questions

Continue learning

01Executive Summary

02Overview

03Learning Objectives

04Definitions

05Core Concepts

The idempotent produced by a one-sided inverse

Two sources of the property

Finite length

No infinite orthogonal idempotents

Why the radical is transparent

06Key Results

07Proof Techniques and Method

08Worked Example

A ring that fails, and why it is not semilocal

Bass' Theorem in a small semilocal ring

09Comparison and Classification

10Relationship Map

11Applications and Industry Use

12Failure Modes and Common Mistakes

13Historical Notes and Lessons Learned

14Quick Reference

15Frequently Asked Questions

16Related KEVOS Topics

17References

18AI Suggested Questions

Continue learning

Executive Summary

Overview

Learning Objectives

Definitions

Core Concepts

Key Results

Proof Techniques and Method

Worked Example

Comparison and Classification

Relationship Map

Applications and Industry Use

Failure Modes and Common Mistakes

Historical Notes and Lessons Learned

Quick Reference

Frequently Asked Questions

Related KEVOS Topics

References

AI Suggested Questions